The GATE Grind

GATE 2024 DA – Question 37

Calculus and Optimization · Limits, continuity and differentiability · 2 marks · Multiple choice

Let $f: \mathbb{R} \to \mathbb{R}$ be a function.

$f(x) = \begin{cases} -x, & \text{if } x < -2 \\ ax^2 + bx + c, & \text{if } x \in [-2, 2] \\ x, & \text{if } x > 2 \end{cases}$

Which ONE of the following choices gives the values of $a, b, c$ that make the function $f$ continuous and differentiable?

  1. $a = \frac{1}{4}, b = 0, c = 1$
  2. $a = \frac{1}{2}, b = 0, c = 0$
  3. $a = 0, b = 0, c = 0$
  4. $a = 1, b = 1, c = -4$

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Show answer and explanation

Correct answer: (A) $a = \frac{1}{4}, b = 0, c = 1$

Explanation

At $x = 2$ the pieces must match in value, $4a + 2b + c = 2$, and in slope, $4a + b = 1$. At $x = -2$ they must match in value, $4a - 2b + c = 2$, and in slope, $-4a + b = -1$. Adding the two slope equations gives $2b = 0$, so $b = 0$ and $a = \frac{1}{4}$. Then $c = 2 - 1 = 1$.